The commutation relations between the generalized Pauli operators of N-qudits (i. e., Np-level quantum systems),
and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical
pattern. One may identify vertices/points with the operators so that edges/lines join commuting pairs of them
to form the so-called Pauli graph PpN . As per two-qubits (p = 2, N = 2) all basic properties and partitionings
of this graph are embodied in the geometry of the symplectic generalized quadrangle of order two, W(2). The
structure of the two-qutrit (p = 3, N = 2) graph is more involved; here it turns out more convenient to deal
with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the geometry
of generalized quadrangle Q(4, 3), the dual of W(3). Finally, the generalized adjacency graph for multiple
(N > 3) qubits/qutrits is shown to follow from symplectic polar spaces of order two/three. The relevance of
these mathematical concepts to mutually unbiased bases and to quantum entanglement is also highlighted in some detail.